Starting from the works of Nakajima and Grojnowski, moduli spaces and stacks of sheaves on surfaces represent wonderful tools for the study of vertex and quantum algebras and their representations from a geometric point of view. For example, Schiffmann and Vasserot proved that the equivariant K-theory of the stack of zero-dimensional sheaves on $\mathbb C^2$ has an associative algebra structure and is isomorphic to the positive part of quantum toroidal algebra of type $\mathfrak{gl}(1)$; moreover, it acts on the equivariant K-theory of the Hilbert scheme of points on $\mathbb{C}^2$. Their result can be seen as a K-theoretic version of Nakajima-Grojnowski cohomological result for Hilbert schemes of points.

In the present talk, I would like to describe a new conjectural approach to the study of quantum toroidal algebras of type $\mathfrak{gl}(k)$ based on the study of algebra structures on the K-theory of the stacks of torsion sheaves over other noncompact surfaces (e.g. the stack of sheaves on the minimal resolution of the Du-Val singularity $\mathbb{C}^2/\mathbb{Z}_k$​, supported at an exceptional curve)​. (This is a work in progress with Olivier Schiffmann.)

The topics of this session are as follows: ---Heat kernel and its asymptotic expansion, Gilkey's formula, Mackean-Singer formula, ---The index problem for elliptic PDE's, characteristic classes via Chern-Weil theory.

The aim of the talk is to show in two (closed) contexts the importance of the knowledge of the set of decomposed places in an infinite extension: (i) for the mu-invariant; (ii) for the exponent of the class group along a p-tower.

This is joint work with F. Hajir (U. Mass.)